We present the first results of the ALMA Fornax Cluster Survey (AlFoCS): a complete ALMA survey of all members of the Fornax galaxy cluster that were detected in HI or in the far infrared with Herschel. The sample consists of a wide variety of galaxy types, ranging from giant ellipticals to spiral galaxies and dwarfs, located in all (projected) areas of the cluster. It spans a mass range of 10^(~8.5 - 11) M_Sun. The CO(1-0) line was targeted as a tracer for the cold molecular gas, along with the associated 3 mm continuum. CO was detected in 15 of the 30 galaxies observed. All 8 detected galaxies with stellar masses below 3x10^9 M_Sun have disturbed molecular gas reservoirs, only 6 galaxies are regular/undisturbed. This implies that Fornax is still a very active environment, having a significant impact on its members. Both detections and non-detections occur at all projected locations in the cluster. Based on visual inspection, and the detection of molecular gas tails in alignment with the direction of the cluster centre, in some cases ram pressure stripping is a possible candidate for disturbing the molecular gas morphologies and kinematics. Derived gas fractions in almost all galaxies are lower than expected for field objects with the same mass, especially for the galaxies with disturbed molecular gas, with differences of sometimes more than an order of magnitude. The detection of these disturbed molecular gas reservoirs reveals the importance of the cluster environment for even the tightly bound molecular gas phase.